Question: Wei is standing in wavy water and notices the depth of the waves varies in a periodic way that can be modeled by a trigonometric function. He starts a stopwatch to time the waves. After $1.1$ seconds, and then again every $3$ seconds, the water just touches his knees. Between peaks, the water recedes to his ankles. Wei's ankles are $12\text{ cm}$ off the ocean floor, and his knees are $55\text{ cm}$ off the ocean floor. Find the formula of the trigonometric function that models the depth $D$ of the water $t$ seconds after Wei starts the stopwatch. Define the function using radians. $ D(t) = $
Explanation: Let's start by writing a formula for the height of the water $u$ seconds after its peak. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. That way, since the water is at its peak at time $u = 0$, let's use a cosine function to model its distance, since cosine functions also reach a peak at $u = 0$. The depth of the water has period $3$ seconds. Its midline is halfway between its maximum of $55 \text{ cm}$ and its minimum of $12\text{ cm}$, or $\dfrac{55 + 12}{2} = 33.5$ Its amplitude is half the difference between its maximum of $55 \text{ cm}$ and its minimum of $12 \text{ cm}$, or $\dfrac{55 - 12}{2} = 21.5$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{3}{2\pi}}$, stretch it vertically by a factor of ${21.5}$, and move it up ${33.5}$ units: $ D(u) = {21.5}\cos\left({\dfrac{2\pi}{3}}u\right) + {33.5}$ Since the water reaches its peak $1.1$ seconds after the stopwatch is started, $t$ seconds after the stopwatch is started is $t - 1.1$ seconds after that peak, so $u = t - 1.1$ : $ D(t) = {21.5}\cos\left({\dfrac{2\pi}{3}}(t-1.1)\right) + {33.5}$ The function $ D(t) = {21.5}\cos\left({\dfrac{2\pi}{3}}(t-1.1)\right) + {33.5}$ has period $3$, amplitude $21.5$ and midline $y = 33.5$, and it reaches its peak at time $t = 1.1$, so it's a good model of the depth of the water.